Multiscale Brazil nut effects in bioturbated sediment

Size segregation in granular materials is a universal phenomenon popularly known as the Brazil nut effect (BNE), from the tendency of larger nuts to end on the top of a shaken container. In nature, fast granular flows bear many similarities with well-studied mixing processes. Instead, much slower phenomena, such as the accumulation of ferromanganese nodules (FN) on the seafloor, have been attributed to the BNE but remain essentially unexplained. Here we document, for the first time, the BNE on sub-millimetre particles in pelagic sediment and propose a size segregation model for the surface mixed layer of bioturbated sediments. Our model explains the size distribution of FN seeds, pointing to a uniform segregation mechanism over sizes ranging from < 1 mm to > 1 cm, which does not depend on selective ingestion by feeding organisms. In addition to explaining FN nucleation, our model has important implications for microfossil dating and the mechanism underlying sedimentary records of the Earth’s magnetic field.


Scientific Reports
| (2022) 12:11450 | https://doi.org/10.1038/s41598-022-14276-w www.nature.com/scientificreports/ finite waiting time and jump length distributions 23 , the concentration C of a conservative tracer (in our case, microtektites) inside a SML where each volume element undergoes sufficient (i.e., ~ 25) bioturbation events 24 before being definitively buried, is governed by a simplified version of the diffusion-advection equation obtained by neglecting the porosity gradient 25,26 : where t and z are the time and the depth below the sediment-water interface, respectively, D t is the diffusion coefficient of the tracer particles, v b the bulk burial velocity, and v t an additional upward velocity of the tracer particles, due for instance to bio-advection or size segregation (Fig. 2c). The upper boundary condition (v b − v t )C − D t ∂ z C = F t /ϕ s ρ s at z = 0, where ϕ s is the volume fraction of solids and ρ s their density, is controlled by the incoming tracer flux F t . A microtektite input event is then described by F t = � t δ(t) where δ(t) is the Dirac impulse, and t the microtektite fluence. Bioturbation intensity declines with depth, so that D t and v t are unknown functions of z. In practice, different depth-dependent diffusion models yield similar fits to experimental data 25,27 , which means that the SML can be represented by an equivalent homogeneous layer with thickness L and constant D t , v b and v t . Solution of Eq. (1) with C(0, z) = δ(z) yields the impulse response I(t) = C(t, L) of the system, which can be converted to a depthdependent concentration profile over z > L using the age model of the sediment (Fig. 2d). The microscopic  particles (e.g., a microtektite fragment) perform a biased random walk starting from the sediment surface, until disturbances cease below the SML. (c) Under certain conditions, the random walk in (b) is governed by a diffusion-advection equation, where the advective velocity is the sum of burial, bioadvective, and size segregation velocities. (d) Depth-dependent concentration C of tracer particles that are small enough to get buried (e.g., microtektite fragments, solid line) and of ferromanganese nodules large enough to stay indefinitely in the top part or the SML (dashed line). The curve below the SML indicates the microtektite distribution inside the historical layer resulting from an instantaneous deposition process. www.nature.com/scientificreports/ equivalent to the impulse response is a Wiener process with constant drift, starting at (t, z) = (0, 0) and ending at (t, z) = (t L , L) , where t L is the transit (or escape) time 28 with probability density function I(t) . The age T of particles found at depth z > L is a stochastic variable related to t L by T = t L + t b , where t b is the burial time from the bottom of the SML derived from the age model. The stochasticity of T is an important factor affecting single specimen dating 29 . Solution examples (Fig. 3a) show how v t increases the time needed to cross the SML, due to the reduced or inverted tracer velocity v b − v t . The mean transit time t L , defined as the expectation of I(t) , diverges above a critical v t /v b threshold (Fig. 3b). This threshold is close to 1 when advection is the dominant transport mechanism of the tracer particles across the SML. Diffusion ensures a non-negligible probability to escape the SML even if v t > v b , yielding a higher v t /v b threshold that depends on the inverse Péclet number G = D s /Lv b of the bulk sediment, where D s is the bulk diffusion coefficient. In all cases, I(t) becomes dramatically skewed as the threshold is approached, converging to a uniform distribution over t > 0. This means that size segregation tends to redistribute large particles above the stratigraphic depth corresponding to their deposition age. The grain size dependence of I(t) has obvious consequences for dating. While G affects the skewness of I(t) , and thus the stratigraphic age of individual particles, but not the mean age 26 -since �t L � = L/v b for v t = 0-size segregation increases the apparent age of larger particles with respect to the bulk, individually and on average, until a meaningful stratigraphic relation is lost.

Microtektite profiles
The expected concentration of microtektites belonging to a given size interval [s 1 , s 2 ] is governed by the model function where g t is the empirical grain size distribution determined from microtektite counts over all depths (Supplementary Figure S1), z 0 the depth in sediment corresponding to the time of the deposition event, and I the impulse response obtained from the solution of Eq. (1). While v b is derived from the age model of the sediment core, t , D t , L , and v t must be determined by fitting microtektite profiles for different size classes. Because conventional (2) cannot be entirely significant, so that additional constraints need to be applied to the size dependencies of D t and v t . Segregation of large particles in the SML can be driven either directly by the BNE, or indirectly by biogenic graded bedding resulting from the selective transport of finer particles through ingestion [31][32][33] , burrow lining 34 , infills 35 , and resuspension 36 . While graded bedding has been observed in sediments dominated by single benthic organisms, it is not a typical feature of regularly deposited sediment 37,38 . Furthermore, size segregation resulting from graded bedding is not expected to depend on particle size above the maximum dimension of ingestible particles, while, as shown later, this dependence is required to explain the formation of FN. Plastic deformation of sediment around burrowing benthic organisms 39,40 is a possible BNE driving mechanism, because the deformation field includes a vertical gradient of horizontal displacement around burrow tips, which is analogous to the horizontal shearing used in many BNE experiments 6,41 . In this case, both the diffusivity and the segregation velocity are proportional to the shear rate 6,42 . Size segregation might also be driven by microbial-induced bubble formation in organic-rich sediments 43 .
Experiments with sorted glass beads 44 , which share with microtektites the lack of preferential ingestion by feeding organism 31 , indicate that the size dependence of tracer diffusivity is governed by a power law of the form D t ∝ s −q with q ≈ 0.52. Percolation of smaller particles through random media also displays a power-law dependence on particle size 45 . Accordingly, we assume that the diffusion of large tracer particles with size s in a sediment with mean grain size s 0 is given by D t = D s (s/s 0 ) −q . Granular mixing experiments show that the advective velocity of large grains is proportional to s/s 0 − ψ c , where ψ c ≈ 2.8 is a critical size ratio threshold in binary mixtures 4,8 . Therefore, we model the segregation velocity as v t = (s/s 0 − ψ c )β 0 D s , with β 0 being an unknown coefficient that expresses the segregation efficiency of bioturbation. Most granular mixing studies have been performed with cohesionless particles, which are a poor analogue to fine-grained sediment. Experiments with wetted grains show that cohesiveness tends to reduce size segregation 46 , as long a clumping is prevented, although this effect is much less pronounced in case of non-spherical particles 47 . Cohesive forces tend to suppress the ability of small particles to infiltrate voids, as they cannot freely fall. However, the biasing effect of gravity, which is the primary cause of size segregation, does not cease. A similar effect has been observed for the torque experienced by magnetic particles in a cohesive, bioturbated sediment, in presence of a weak magnetic field 48,49 . In this case, the resulting magnetic alignment was found to be proportional to the ratio between the magnetic torque and the torques that resist particle rotation. With these considerations in mind, effects of sediment cohesiveness are entirely accounted by β 0 . A Poisson regression model has been used to fit microtektite counts (Fig. 4) for three size classes, using the above models for D t (s) and v t (s) . Model residuals are generally compatible with counting uncertainties estimated with bootstrapping, up to few exceptions that might be explained by sediment heterogeneity. The size segregation parameters β 0 and q are both significantly different from zero at a > 99.4% confidence level ( Table 1). Estimates of D s and L are comprised within the typical ranges obtained from radioactive tracers for similar sediments 50 . The power-law exponent q ≈ 0.25 for the size dependence of D s (Table 1) is smaller than the value obtained by Wheatcroft 44 for 10-300 µm glass beads, possibly because most microtektites are too large to be ingested.

Stratigraphic, environmental, and paleomagnetic implications
Tektite profiles illustrate how size segregation offsets the age distribution of buried objects (solid lines in Fig. 4a), relative to that of regular sediment particles (dashed lines in Fig. 4a). The predicted mean age offsets for the three size classes of Fig. 4 (Fig. 3). For this reason, sediments with low deposition rates are expected to be particularly prone to age offsets caused by size segregation. For instance, decreasing v b to 1.5 cm/kyr for a sediment with same properties as MD90-0961 would increase the age offset of a 0.5 mm object from ~ 0.73 to ~ 10 kyr. Large positive offsets caused by the BNE can explain foraminifera age differences that cannot arise from selective dissolution alone 21,51,52 . On the other hand, size segregation is less sensitive to changes of the diffusion coefficient: for instance, doubling D s increases the age offset of the 0.5 mm object of the above example to ~ 1.7 kyr, because the resulting increase of v t is partially compensated by a decrease of Pe. In case of a stationary flux of large tracer particles, the BNE produces a concentration gradient within the SML with a similar dependence on v t /v b as t L (Supplementary Figure S2): this is because conservation of the vertical tracer flux requires a decrease of the net burial velocity to be compensated by a higher concentration. Hence, the interpretation of foraminiferal concentration variations within the SML might be biased by the BNE. For instance, the concentration of G. bulloides tests in sediments of the Oman margin 53,54 , which have been used to reconstruct the Indian summer monsoon during the last ~ 2000 years, might increase by 9-40% in the uppermost ~ 6 cm, if the same segregation parameters of Table 1 are assumed along with bioturbation data representative of the oxygen minimum zone in the northwest Arabian Sea (i.e., D s ≈ 150 cm 2 /kyr, L ≈ 6 cm, and v b = 3-20 cm/kyr 54,55 ).
Size segregation at sub-millimetre scales has important paleomagnetic implications because it requires a reorganization of the sediment microstructure assimilable with a truly diffusive process, which causes a reorientation of magnetic carriers. While non-local transport in the SML is hardly distinguishable from true diffusion 23 , the two processes affect sedimentary records of the Earth magnetic field in a drastically different manner. Upward conveyor belt transport removes material at depth and redeposits it on the sediment surface, where a so-called detrital remanent magnetization (DRM) is acquired by partial alignment of suspended particles in the magnetic field. Buried sediment, which is not affected by bioturbation in this model, would carry an intact DRM coeval with deposition age 56 . Local disruption of the sediment structure, on the other hand, erases the existing DRM and replaces it with a post-depositional magnetization (PDRM) younger than the age of deposition. Conventional PDRM models assume that this magnetization is acquired below the SML during early diagenesis 57 ; however, laboratory experiments have shown that PDRM acquisition can be driven by bioturbation through the rotational component of diffusion 48,49 . Diffusive sediment mixing introduces a delay of the order of L/v b in magnetostratigraphic records not affected by diagenesis. This delay is compatible with observed offsets between magnetic mineral and 10 Be records of the Matuyama-Brunhes field reversal 58 , if appropriated estimates of L for marine sediments 50 are used.

Implications for ferromanganese nodules
If FN nucleation and microtektite segregation have the same origin, segregation parameters estimated from microtektite profiles can be used to predict the minimum size of FN nuclei, which is 1-5 mm 18 . A simple growth model assumes that the size s(T) = s n + γ T of a FN of age T increases linearly in time from the initial nucleus size s n at a constant growth rate γ ≈ 1-5 mm/Myr 59 . In case of a stationary flux F 0 of seeds with initial grain size distribution n 0 (s) , the size distribution n(s) of growing seeds at the sediment-water interface is given by where r(s) is the ratio between the tracer concentrations at z = L and z = 0, respectively, obtained from the steadystate solution of Eq. (1). Solutions of Eq. (3) with the size segregation parameters of Table 1 and SML properties typical of FN fields at a water depth of ~ 4000 m 50,60 predict minimum seed sizes of ~ 2-3 mm (Fig. 5), which are comparable with observed ones 18 . Under these conditions, an increase of v b from 0.5 to 0.8 cm/kyr, a decrease of D s from 22 to 8 cm 2 /kyr, or a decrease of β 0 from 0.075 to 0.064 m −1 are sufficient to suppress the growth Table 1. Poisson regression models for two null hypotheses H 0 (fixed parameters in parentheses) and the size segregation hypothesis H 1 . The p value represents the rejection probability of H 0 = H 1 , obtained from the test statistics −2ln of the likelihood-ratio test. Error estimates for H 1 parameters correspond to the standard deviation of bootstrapped simulation of microtektite counts. Fixed parameters derived from the age model and from physical properties are v b = 3.8 cm/kyr, ϕ s = 0.6, ρ s = 2.6 g/cm 3 , s 0 = 10 µm, ψ c = 3, and ϕ s (z 0 )/ϕ s (L) = 1.33.
Parameter H 0 ( β 0 = q = 0) H 0 ( β 0 = 0) H 1 ( β 0 > 0, q > 0)  17 . Finally, a linear dependence of v t on size up to at least ~ 4 mm, well above the limit of selective transport by benthic organisms, is required to reduce the sinking probability of growing FN nuclei so that they can grow by several cm (dashed line in Fig. 5). This demonstrates that FN are kept at the sediment surface by the BNE, rather than other mechanisms like the selective removal of fine-grained sediment by bottom currents 17 .

Conclusions
We report, for the first time, the size segregation of 0.05-0.9 mm microtektite fragments in a pelagic sediment from the Indian Ocean. The depth distribution of these fragments can be explained by a bioturbation-driven BNE in the SML. As a result, large particles experience an upwardly directed segregation velocity v t relative to the bulk sediment, which increases linearly with particle size. Above a sediment-dependent size threshold (e.g., ~ 1 mm in core MD90-0961), v t exceeds the burial velocity, and the probability of burial below the SML becomes small. This has two important consequences: (1) buried objects that are much larger than the mean grain size of sediment, such as microfossils, are significantly older than their stratigraphic age and tend to lose any relation with stratigraphy for sizes above the v t = v b threshold, and (2) > 1 mm particles tend to remain on the sediment surface for long times, serving as seeds for the growth of FN under favourable conditions. In the latter case, continuous growth further decreases the burial probability, explaining the scarcity of buried nodules. A single empirical model for the size dependence of segregation velocity and diffusivity, derived from experiments on granular mixing, explains our microtektite counting results and correctly predicts the minimum size of FN seeds, despite the > 10 orders of magnitude difference between bioturbation and laboratory time scales. Plastic deformation of sediment associated with burrowing is the most likely BNE driving mechanism. The BNE has important implications for the fundamental understanding of phenomena that depends on sediment micromechanics, such as FN growth and paleomagnetic records, and for the interpretation of foraminifera ages and concentration variations. Large positive age offsets caused by the BNE can explain foraminifera age differences that cannot arise from selective dissolution alone. Furthermore, the BNE produces a concentration gradient within the SML, which might affect the interpretation of recent climatic variations. The effect of physical sediment properties like cohesiveness on size segregation needs to be investigated to assess the role played by the BNE in the redistribution of large particles in sediment, beyond the single example presented here.

Methods
Microtektite counting. Core MD90-0961 (5°03.71′ N, 73°52.57′ E) was collected during the SEYMAMA research cruise of the R/V Marion Dufresne in 1990. The 45-m long core was retrieved on the eastern margin of the Chagos-Maldive-Laccadive Ridge at a water depth of 2450 m and is composed of calcareous nannofossil ooze with abundant foraminifera. Typical microtektite concentrations amount to few counts per sample (~ 3 g); therefore, counts from three sampling campaigns (Supplementary Tables S1, S2, S3) have been gathered into three size classes with a total of 137, 287, and 49 counts, respectively (Table 2). Sediment preparation details are given in the Supplementary Information.